
OUR 



•CALENDAR 



-o-^3*@^ 



BY 



REV. GEORGE N. PACKER, 



Wallsbarp, Fa, 



OUR 

CALENDAR 



The Juliaii Calendar* and its errors. 
How corrected by the Grregorian which 
is now in use almost throughout the civ- 
ilized World. Also Rules for finding 
the Dominical Letter and the day of 
the week of any event, in any Year, 
from the commencement of the Chris- 
tian Era to the Year of our Lord 4000. 



ILLUSTRATED BY VJ\LUJ{BLE TABLES Afi/D Cf/A/fTS 

BY 

REV. GEORGE NICHOLS PACKER. 



WELLSBORO, PA. 

REPUBLICAN ADVOCATE PRINT. Zl "Z.Q f ?') 

1890. X^HingTOM. 

**3 






THE LIBRAEY 

OF CONGRESS 

WASHINGTON 



Entered according to Act of Congress, in the year 1890, 

By Rev. George Nichols Packer, 

In the Office of the Librarian of Congress, at Washington, D. C. 



TO 

HON. HENRY W. WILLIAMS, 

JUSTICE OF THE SUPREME COURT 
OF 

PENNSYLVANIA, 

WHOM 

I HAVE FOUND A TRUE FRIEND IN POVERTY AND IN SICKNESS, 

AND 

FROM WHOM I HAVE RECEIVED WORDS OF ENCOURAGEMENT 

AND COMFORT DURING MANY YEARS OF ADVERSITY, 

AND AT 
WHOSE SUGGESTION THIS LITTLE VOLUME HAS BEEN WRITTEN, 

AND BY 

WHOSE ASSISTANCE IT IS NOW PUBLISHED, 
THIS 

HUMBLE VOLUME IS DEDICATED 

AS A 

TRIBUTE OF RESPECT 

BY THE 

AUTHOR. 



PffKFJlGE 



MANY years ago while engaged in teaching, the writer of 
this little volume was in the habit of bringing to the at- 
tention of his pupils a few simple rules for finding the dominical 
letter and day of the week of any given event within the past and 
the present centuries ; further than this he gave the subject no spe- 
cial attention. 

A few years ago having occasion to learn the day of the week 
of certain events that were transpiring at regular intervals on the 
same day of the same month, but in different years, he was led to 
investigate the subject more thoroughly, so that he is now able 
to give rules for finding the dominical letter and the day of the 
week of any event that* has transpired or will transpire, from 
the commencement of the Christian era to the year of our Lord 
4,000, and to explain the principles on which these rules rest. 
When the investigations were entered upon he had no thought of 
writing a book ; but having been laid aside from active labor by 
ill health, he found relief from the despondency in which sickness 
and poverty plunged him by pursuing the study of the calendar, 
ts history, and the method of disposing of the fraction of a day 
found in the time required for the revolutio n of the Earth in its 
orbit about the Sun. 

He became so much interested in the study of this subject that 
he frequently spoke of it to friends and acquaintances whom he 
met. On one occasion, while speaking to Hon. H. W. Williams 
about some of the curious results of the process by which the co- 
incidence of the solar and the civil year is preserved, it was suggest- 
ed to him that he should put the story of the calendar, its correc- 
tion by Gregory, and the theory and results of intercalation, in 



6 

writing. It was urged that this would give increased interest to 
the study, help the writer to forget his pains, and probably en- 
able him to realize a little money from the sale of his work to meet 
pressing wants. Acting upon this suggestion an effort has been 
made to put into this little volume some of the most interesting 
facts relating to the origin, condition, and practical operation of 
the calendar now in use ; together with rules for rinding the day 
of the week on which any given day of any month has fallen or 
will fall during four, thousand years from the beginning of our era 
The writer does not claim absolute originality for all that ap- 
pears in the following pages ; on the contrary, he has made free 
use of all the materials that came within his reach relating to the 
history of the calendar and the work of its correction by Gregory. 
These materials together with his own calculations he has arrang- 
ed in accordance with a plan of his own devising, so that the out- 
line and the execution of the work may be truly said to be origi- 
nal. Of its value the world must judge. It has been prepared in 
weakness of body and in suffering, which have been to some ex- 
tent relieved by the mental occupation thus afforded, but which 
may have nevertheless left their impress on the work. But let it 
be read before pronouncing judgment upon it. Cicero could infer 
the littleness of the Hebrew God from the smallness of the territory 
He had given his people. To whom Kitto replies : ' ' The interest 
and importance of a country arise, not from its territorial extent, 
but from the men who form its living soul ; from its institutions 
bearing the impress of mind and spirit, and from the events 
which grow out of the character and condition of its inhabitants." 
So the value of a book does not consist in the size and number of 
its pages, but from the knowledge that may be gained by it 
perusal. The Author. 



CONTENTS. 



PART FIRST. 

DEFINITIONS— HISTORY. 



Pages. 
Chapter I. — Definitions 9-10 

Chapter II. — History of the divisions of time, and the old 

Roman calendar 10-15 

Chapter III —History of the reformation of the calendar 

by Julius Csesar 16-17 

Chapter IV —History of the reformation of the Julian cal- 
endar by Pope Gregory XIII 18-25 



PART SECOND. 

MATHEMATICAL. 



Chapter I.— Errors of the Julian calendar . . . 26 28 

Chapter II. — Errors of the Gregorian calendar 28-29 

Chapter III— Dominical letter 30-35 

Chapter IV. —Rule for finding the dominical letter 36-41 

Chapter V -Rule for finding the day of the week of any 

given date, for both Old and New Styles 42-51 

Chapter VI.— A simple method for finding the day of the 
week of events, which occur qudrennially ; the inaugural 
of the Presidents, the day of the week on which they have 
occurred and on which they will occur for the next one 

hundred years 52-54 

Some peculiarities concerning events which fall on the 29th 

of February 55-59 

Appendix 60-68 



OUR CALENDAR 



PART FIRST. 



DE FIN [TION 3. HISTORY. 



CHAPTER I. 



DEFINITIONS. 



a — A Calendar is a method of distributing time 
into certain periods adapted to the purposes of civil 
life, as hours, days, weeks, months, years, eic. 

b — An hour is the subdivision of the day into 
twenty-four equal parts. 

c — The true solar day is the interval of time 
which elapses between two consecutive returns of 
the same terrestrial meridian to the Sun. the mean 
length of which is twenty- four hours. 

d — The week is a period of seven days having no 
reference whatever to the celestial motions, a circum- 
stance to which it owes its unalterable uniformity. 

9 



10 

e — The month is usually employed to denote an 
arbitrary number of days approaching a twelfth part 
of a year, and has retained its place in the calendar 
of all nations. 

f — The year is either astronomical or civil. The 
solar astronomical year is the period of time in 
which the Earth performs a revolution in its orbit 
about the Sun or passes from any point of the eclip- 
tic to the same point again, and consists of 365 days, 
5 hours, 48 minutes and 49.62 seconds of mean solar 
time. Appendix A. 

6 —The civil year is that which is employed in 
chronology, and varies among different nations both 
in respect of the seasons at which it commences and 
of its subdivisions. 



CHAPTER II. 



HISTORY OF THE DIVISIONS OF TIME, AISTD THE OLD 
ROMAN CALENDAR. 



Day— The subdivision of the dayinto twenty-four 
parts or hours has prevailed since the remotest ages, 
though different nations have not agreed either with 
respect to the epoch of its commencement or the man- 
ner of distributing the hours. Europeans in general, 



11 

like the ancient Egyptians, place the commencement 
of the civil day at midnight ; and reckon twelve 
morning hours from midnight to midday and twelve 
evening hours from midday to midnight. Astrono- 
mers after the example of Ptolemy, regard the day 
as commencing with the Sun's culmination, or noon, 
and find it most convenient for the purpose of com- 
putation to reckon through the whole twenty-four 
hours. Hipparchus reckoned the twenty-four hours 
from midnight to midnight. 

Week — Although the week did not enter into the 
calendar of the Greeks, and was not introduced at 
Rome till after the reign of Theodosius, A. D. 392, it 
has been employed from time immemorial in almost 
all Eastern countries ; and as it forms neither an ali- 
quot part of a year nor of the lunar months, those who 
reject the Mosaic recital will be at a loss to assign to 
it an origin having much semblance of probability. 
In the Egyptian astronomy the order of the planets, 
beginning with the most remote, is Saturn, Jupiter, 
Mars, the Sun, Venus, Mercury, the Moon. Now, 
the day being divided into twenty four hours, each 
hour was consecrated to aparticnlar planet, namely: 
One to Saturn, the following to Jupiter, third, to 
Mars, and so on according to the above order ; and the 
day received the name of the planet which presided 
over its first hour. If, then, the first hour of a day 
wa? consecrated to Saturn, that planet would also 
have the 8th, the 15th, and the 22d hours ; the 23d 



12 
would fall to Jupiter, the 24th to Mais, and the 25th 

or the first hour of the second day would belong to 
the Sun. In like manner the first hour of the 3d 
day would fall to the Moon, the first hour of the 4th 
to Mars, of the 5th to Mercury, of the 6th to Jupiter, 
and the 7th to Venus. * 

The cycle being completed, the first hour of the 8th 
day would again return to Saturn and all the others 
succeed in the same order. It will be seen by the 
table at the close of this chapter, and it is also re- 
corded by Dio Cassius, of the 2d Century, that the 
Egyptian week commenced with Saturday. On their 
flight from Egypt the Jews, from hatred to their an- 
cient oppressors, made Saturday the last day of the 
week. It is stated that the ancient Saxons borrow- 
ed the week from some Eastern nation, and substi- 
tuted the names of their own divinities for those of 
the gods of Greece. The names of the days are 
here given in Latin, Saxon, and English. It will be 
seen that the English names of the days are derived 
from the Saxon. 



LATIN. 

Dies Solis. 
Dies Lance- 
Dies Martis. 
Dies Mercurii. 
Dies Jovis. 
Dies Veneris. 
Dies Saturni, 



SAXON. 

Sun's Day. 
Moon's Day. 
Tiw's Day. 
Woden's Day. 
Thor'sDay. 
Friga's Day. 
Seterne's Day. 



ENGLISH. 

Sunday. 

Monday. 

Tuesday .• 

Wednesday. 

Thursday. 

Friday. 

Saturday. 



13 

Month — The ancient Roman year contained but 
ten months and is indicated by the names of the last 
four. September from Septem, seven ; October from 
Octo, eight; November from, Novem, nine; and 
December from Decern, ten; July and August 
were also denominated Quintilis, and Sextilis; from 
Quintus, live; and Sex, six. 

Quintilis was changed to July in honor of Julius 
Caesar, who was born on the 12th of that month 98 
B. C. Sextilis was changed to August by the Roman 
Senate to flatter Augustus on his victories about 8 
B. C. in the reign of Numa Pompilius, about 700 
B. C, two months were added to the year, January 
at the beginning, and February at the end of the 
year. This arrangement continued till 4o0 B. C, 
when the Decemvirs (ten magistrates) changed the 
order placing February after January, making March 
the 3d instead of the 1st month of the Roman year. 

Year — If the civil year correspond with the solar 
the seasons of the year will always come at the same 
period. But if the civil year is supposed to be too 
long (as is the case in the Julian year) the seasons 
will go back proportionately ; but if too short, they 
will advance in the same proportion. Now as the 
ancient Egyptians reckoned thirty days to the 
month invariably ; and to complete the year, added 
five days called supplementary days, their year 

consisted of 365 days. 
They made use of no intercalation, and by losing 



14 
one-fourth of a day every year, the commencement 
of the year went back one day in every period of 
four years, and consequently made a revolution of 
the seasons in 1461 years. Hence 1460 Julian years 
of 365 1-4 days each are equal to 1461 Egyptian years 
of 365 days each. 

The ancient Roman year consisted of 355 days. 
This differed from the solar year by ten whole days 
and a fraction ; but to restore the coincidence, Numa 
ordered an additional or intercalary month to be inser- 
ted every second year between the 23d and 24th of 
February, consisting of twenty-two and twenty- three 
days alternately, so that four years contained 1465 
days and the mean length of the year was conse- 
quently 366 1-4 days, so that the year was then too 
long by one day. 

The year was then reduced to 365 14 days by 
suppressing the intercalation in every period of 
eight years. Had the intercalations been regularly 
made the concurrence of the solar and the civil year 
would have been preserved very nearly. But its 
regulation was left to the pontiffs, who to prolong 
the term of a magistracy or hasten an annual election 
would give to the intercalary month a greater or less 
number of days and consequently the calendar was 
thrown into confusion, so that in the time of Julius 
Caesar there was a discrepancy between the solar 
and the civil year of about three months ; the winter 
months being carried back into the autumn and 
the autumnal into summer. Appendix B. 



15 

A table of the order and the names of the planets 
in the Egyptian astromomy illustrating the origin 
of the names of the days of the week : 



=5 8 


I Jupiter, 
*° j Thursday. 


'Mars, 
01 | Tuesday. 


.Si! 
if 

4 


ll 11 11 

1 ^ 


1 


5 ! 6 


7 


8 


9 


10 


11 


12 13 


14 


15 


16 


17 


18 


19 20 


21 


22 


23 


24 


1 


2 3 


4 


5 


6 


7 


8 


9 10 


11 


12 


13 


14 


15 


16 17 


18 


19 


20 


21 


22 


23 24 ; 


1 


2 


3 


4 


5 ; 


6 ; 7 


8 


9 


10 


11 


12 


13 ; 14 


15 


16 


17 


18 


19 


20 21 


22 


23 


24 


1 


2 


3 4 


5 


6 


7 


8 


9 


10 11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 


1 


2 


3 


4 


5 


6 


7 8 


9 


10 


11 


12 


13 


14 ; 15 


16 


17 


18 


19 


20 


21 i 22 


23 


24 


1 


2 


3 


4 ! 5 


6 


7 


8 


9 


10 


il ; 12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 


1 


2 


3 


4 


5 


6 


7 


8 


9 


10 


11 


12 


13 


14 


15 


16 


17 


18 


19 


20 


21 


22 


23 


24 



16 
CHAPTER III. 



HISTORY OF THE REFORMATION OF THE CALENDAR 
BY JULIUS CESAI*. 



Forty-six years before Christ, Julius Csesar called 
on the astronomers, especially on Sosigones of Alex- 
andria, to assist him in this very desirable under- 
taking. The mean length of the year was fixed at 
365 1-4 days. It was decided that every year should 
consist of 365 days excepting the fourth, which 
should have 366. 

In order to restore the vernal equinox to the 24th 
of March the place it occupied in the time of Nurna, 
two months, together consisting of 67 days were in- 
serted between the last day of November and the 
first day of December of that year. An intercalary 
month of 23 days had already been added to Febru- 
ary of the same year according to the old method, 
so that the first Julian year commenced with the first 
day of January, 45 years before Christ ; and 709 
from the foundation of Rome, making the year A. 
U. C. 708 to consist of the prodigious number of 445 
days. (I. e. 355+23+67=445.) Hence it was called 
by some the year of confusion ; Macrobius said it 
should be named the last year of confusion. 

There was also adopted at the same time a more 
commodius arrangement in the distribution of the 
days through the several months. It was decided 
to give to January, March, May, July, September and 



17 
November each thirty-one days ; and the other 
months thirty, excepting February which in com- 
mon years should have only twenty-nine days, but 
every fourth year thirty ; so that the average length 
of the Julian year was 365 1-4 days. 

A ugustus Csesar interupted this order by taking 
one day from February, reducing it to twenty-eight 
and giving it to August, that the month bearing his 
name should have as many days as July, which 
was named in honor of his great-uncle Julius. In 
order that three months of thirty-one days might 
not come together, September and November were 
reduced to thirty days, and thirty-one given to Oc- 
tober and December. 

In the Julian calandar a day was added to Febr- 
uary every fourth year, (that being the shortest) 
which is called the additional or intercalary day 
ail is inserted in the calendar between the 24th and 
25th of that month. In the ancient Roman calen- 
dar the first day of every month was invariably call- 
ed the calends ; February then having twenty nine 
days, the 25th was the 6th of the calends of March, 
— sexto calendas ; the preceding which was the ad- 
ditional or intercalary day was called bis-sexto calen- 
das,— twice the sixth day. Hence the term bissexti- 
le as applied to every fourth year commonly called 
leap-year. 



18 
CHAPTER IV. 



HISTORY OF THE REFORMATION OF THE JULIAN 
CALENDAR BY POPE GREGORY THE XIII. 



True enough the year, in which Julius Caesar re- 
formed the ancient Roman calendar, was the last 
year of confusion, and the method adopted by him, 
a commodious one, and answered a ver y good purpose 
for a short time but as the years rolled on and century 
after century had passed away, astronomers be- 
gan to discover the discrepancy between the solar 
and the civil year ; that the vernal equinox did not 
occupy the place it occupied in the time of Caesar, 
namely, the 24th of March, but was gradually retro- 
grading towards the beginning of the year, so that 
at the meeting of the council of Nice in 325 it fell 
on the 21st. Appendix C. 

The venerable Bede in the 8th century observed 
that these phenomena took place three or four days 
earlier than at the meeting of that council. Roger 
Bacon in the 13th century wrote a treatise on this 
subject and sent it to the Pope, setting forth the 
errors of the Julian calendar ; the discrepancy at 
this time amounted to seven or eight days. 

Thus the errors of the calendar continued to in- 
crease until 1582, when the vernal equinox fell on 
the 11th instead of the 21st of March. Gregory 



19 

perceiving that the measure (of reforming the cal- 
endar) was likely to confer great eclat on his ponti- 
ficate undertook the ]ong desired reformation ; 
and having found the governments of the principal 
Catholic States ready to adopt his views, he issued 
a brief in the month of March 1582, in which he 
abolished the use of the ancient calendar, and sub- 
stituted that which has since been received in al- 
most all Christian countries under the name of the 
Gregorian calendar or New Style. 

The edict of the Pope took effect in October of 
that year causing the 5th to be called the 15th of 
of that month, thus suppressing ten days, and mak- 
ing the year 1582 to consist of only 355 days. So 
we see that the ten days that had been gained by 
mcorrect computation during the past 1257 years 
were deducted from 1582 restoring the concurrence of 
the solar and the civil year, and consequently the 
vernal equinox to the place it occupied in 325, name- 
ly the 21st of March. 

The Pope was promptly obeyed in Spain, Portu- 
gal, and Italy. The change took place the same 
year in France, by calling the 10th the 20th of De- 
cember. Many other Catholic countries made the 
change the same year, and the Catholic states of Ger- 
many the year following. But most of the Protestant 
countries adhered to the Old Style until after the 
year 1700. Among the last was Great Britain ; she, 
* after having suffered a great deal of inconvenience 



20 
for nearly two hundred years by using a different 
date from the most of Europe, at length, by an act 
of Parliament, fixed on September, 1752, as the 
time for making the much to be desired change, 
which was done by calling the 3d of that month the 
14th, (as the error now amounted to eleven days), 
adopting at the same time the Gregorian rule of in- 
tercalation. 

Russia is the only Christian country that still ad- 
heres to the Old Style, and by using a different date 
from the rest of Europe is now twelve days behind 
the true time. The discrepancy b : ween solai and civil 
time does not effect the day, for, as has already been 
shown, the mean length of the day is twentv-i\ ur 
hours, and is marked by one revolution of the Earth 
upon its axis. 

Nor does it effect the week, for the week is uni- 
formly seven of those days. But it effects the year, 
the month and the day of the month. 

Russia, by adhering to the Old Style, has reckon- 
ed as many days, and as many w r eeks, and events 
have transpired on the same day of the week as they 
have with us who have adopted the New Style ; as 
Christian nations we are observing the same day as 
the Sabbath. 

When it was Tuesday, the 20th day of December, 
1888, in Russia, it was Tuesday, the 1st day of 
January, 1889, in those countries which have adopt- 
ed the New Style. Columbus sailed from Palos, in 
Spain, on Friday, August 3d, 1492, Old Style, which 



21 

was Friday, August 12th, New Style. Washington 
was born on Friday, February 11th, 1732, Old Style, 
which was Friday, February 22d, New Style. 

Now, the difference in styles during the 15th cen 
tury is nine days ; during the 16th and 17th centur- 
ies, ten days; the 18th century, eleven days, and 
the 19th, twelve days. In regard to the sailing of 
Columbus, the change is made by suppressing nine 
days calling the 3d the 12th of August, In regard 
to the birth of Washington, the change is effected 
by suppressing eleven days, calling the 11th of Feb- 
ruary the 22d. As regards Russia, she could have 
made the change last year by calling the 20th of De- 
cember, 1888, the 1st day of January, 1889, thereby 
suppressing twelve days and making the year 1888 
to consist of only 354 days, and the month of Decem- 
ber twenty days. The methods of computation, both 
old and new styles, will be explained in another 
chapter. 

To persons unacquainted with astronomy the dif- 
ference between Old and New Styles would probably 
be better understood by the diagram on the 23d 
page. The figure represents the ecliptic, which is 
the apparent path of the sun, or the real path of the 
Earth as seen from the sun, in her annual or yearly 
revolution around the sun in the order of the months 
as marked on the ecliptic. 

Attention is called to four points on the ecliptic, 
namely, the vernal equinox, the autumnal equinox, 
the winter solstice and the summer solstice. These 



22 
occur, in the order given above, on the 21st of March, 
the 21st of September, the 21st of December and the 
21st of June. It has already been stated that if the 
civil year correspond with the solar, the seasons of 
the year will always come at the same period. Julius 
Caesar found the ancient Roman year in advance of 
the solar ; Gregory found the Julian, behind the 
solar year. So one reforms the calendar by inter- 
calation, the other by suppression. i\ppendix D. 

Caesar restored the coincidence of the solar and 
the civil year, but failed to retain it by allowing 
what probably appeared to him at the time a trifl- 
ing error in his calendar. The error which was 
11 min:ites and 10.38 seconds every year was hardly 
perceptible for a short period, but still amounted to 
three days every 400 yea v s. Hence the necessity in 
1582 of reforming the reformed calendar of Julius 
Caesar to restore the coincidence. Appendix E. 

From the meeting of the Council of Nice, in 325, 
to 1582, a period of 1257 years, there was found to 
be an error in the Julian calendar of ten days. Now, 
in 1257 years the Earth performs 1257 annual and 
459,109 daily revolutions, after w r hich the vernal 
equinox was found to occur on the 21st of March, 
true or solar time ; thus concurring with the vernal 
equinox of 325. But the erroneous Julian calendar 
would make the Earth perform 459,119 revolutions 
to complete the 1257 years ; a discrepancy of ten 
days, making the vernal equinox to fall on the 11th. 
instead of the 21st. It will be seen by diagram that 



23 



21st. 




u *noistijnoo jo md£ 
^SBjaq^,, pn-e 's£ep Qff=LQ-\-$z-\-9Q2 S I WW t 
v 's^p c^ jo isisaoo o; xea^ ^q^ Sappsui 's£ep 
vg\ 06 SuT^xBo.ia^ui iCq ;, q ft 9^ xeseeo snq / 
^ \ -rif .£q pamioja^ 'jBpnaj'eo tremor aqx /^ 



* 



Y> 



eoi^sjog -cctng 



**si3 



24 
ten days were deducted from October in 1582 mak- 
ing it a short month consisting of only twenty-one 
days. 

The discrepancy between the Julian and Gregor- 
ian calander amounts to thirty days in 4000 years ; 
three months in 12,175 years. Hence in 12,175 years 
the equinoxes would take the place of the solstices, 
and the solstices the place of the equinoxes. In 24,- 
350 years, the vernal equinox would take the place 
of the autumnal, and the winter solstice the place of 
the summer solstice. 

And in 43,700 years according to the Julian rule 
of intercalation there would be gained n arly 30"' 1-4 
days, or one entire revolution of the Earth, ^o, to 
restore the concurrence of the Julian and Gregorian 
years, there would have to be suppressed 365 1-4 days, 
calling the 1st day of January 48,699, the 1st day of 
January 48,700. 

Thus would disappear from the Julian calendar 
twelve months or one whole year, it having been di- 
vided among the thousands of the preceding years. 

To make this subject better understood, let us 
suppose the solar year to consist in round numbers 
of 365 days, and the civil year 366. It is evident 
that at the end of the year of 365 days, there would 
still be wanting one day to complete the civil year 
of 366 days, so one day must be added to that year, 
and to every succeeding year, to complete the years 
of 366 days each, which would be the loss of one 



25 

year of 365 days in 365 years. Hence 364 years of 
366 days each are equal to 365 years of 365 days 
each, wanting one day. 

Again, let us suppose the civil year to consist of 
364 days. It is evident that at the end of the sup- 
posed solar year of 365 days, there would be an ad- 
vance or gain of one day in that year and in every 
succeeding year, so that in 365 years there would be 
a gain of 365 days or one whole year. Hence 366 
years of 364 days each are equal to 365 years of 365 
days each, wanting one day. Appendix F. 



PART SECOND. 



MATHEMATICAL. 



CHAPTER I. 



ERRORS OF THE JULIAN CALENDAR. 



It will be necessary in the first place to under- 
stand the difference between the Julian and Grego- 
rian rule of intercalation. It' the number of any 
year be exactly divisible by 4 it is leap-year ; if the 
remainder be 1, it is the first year after leap-year ; if 
2, the second ; if 3, the third ; thus: 

1888-^4=472, no remainder. 

1889-^4=472, remainder, 1. 

1890-^-4=472, remainder, 2. 

1891-^-4=472, remainder, 3. 

1892-^4=473, no remainder, 
and so on, every fourth year being leap-year of 366 
days. 

This is the Julian rale of intercalation, which is 
corrected by the Gregorian, by making every centu- 

26 



27 
rial year, or the year that completes the century a 
common year, if not exactly divisible by 400 ; so that 
only every fourth centurial year is leap-year ; thus 
1,700, 1,800, and 1 900 are common years, but 2,000 
the fourth centurial year is leap-year, and so on. 
By the Julian rule three-fourths of a day is gained 
every century, which in 400 years amounts to three 
days ; this is corrected by the Gregorian, by making 
three consecutive centurial years common years, thus 
suppressing three days in 400 years. 

EULE. 

Multiply the diflerenc between the Julian and the 
solar year by 100 and we have the error in 100 
years. Multiply this product by 4 and we have the 
error in 400 years. Now 400 is the tenth of 4,000 ; 
therefore multiply the last product by 10 and we 
have the error in 4,000 years. Now as the discrep- 
ancy between the Julian and Gregorian year is three 
days in 400 years, making 3-400 of a day every year, 
so by dividing 365 1-4, the number of days in a year, 
by 3-400, we have the time it would take to make 
a revolution of the seasons. 

SOLUTION. 

(365 d, 6 h.)— (365 d, 5 h, 48 m, 49.62 s.) = ( 1:L m > 
10.38 s.) Now (11 m, 10.38 s.)X 100=18 h, 37.3 m, 
the gain in 100 years. This is, reckoned in round 
numbers, 18 hours or three fourths of a day. Now 
(3-4x4)=(lx3)=3 ; the Julian rule gaining three 
days, the Gregorian suppressing three days in 400 



28 
years. (3xl0)=30 the number of days gained by the 
Julian rule in 4,000 years. 365 1-4^3-400=48,700, so 
that in this long period of time, this falling back 3-4 
of a day every century would amount to 365 1-4 
days ; therefore 48,699 Julian years are equal to 48, 
700 Gregorian years. 



CHAPTER II. 



ERRORS OF THE GREGORIAN CALENDAR. 



By reference to the preceding chapter it will be 
seen that there is an error of 37.3 minutes in every 
100 years not corrected by the Gregorian calendar; 
this amounts to only .373 of a minute a year, or one 
day in 3,861 years, and one day and fifty- two min- 
utes in 4,000 years. 

RULE. 

To find how long it would take to gain one day ; 
divide the number of minutes in a day by the deci- 
mal . 373, that being the fraction of a minute gained 
every year. To find how much time would be gain- 
ed in 4,000 years, multiply the decimal .373 by 4,000, 
and you will have the answer in minutes, which 
must be reduced to hours. 



29 

SOLUTION. 

(24 X 60) -=-.373 =3, 861 nearly; hence the error would 
amount to only one day in 3,861 years. 

(.373X4, 000)-60=(24 h, 52 m,)=(l d, h, 52 m,) 
the error in 4,000 years. 

This trifling error in the Gregorian calendar may 
be corrected by suppressing the intercalations in the 
year 4,000 and its multiples 8, 000, 12,000 and 16,000, 
etc., so that it will not amount to a day in 100,000 
years. 

RULE. 

Divide 100,000 by 4,000 and you will have the 
number of intercalations suppressed in 100,000 years. 
Multiply 1 d, 52 m, (that being the errors in 4,000 
years,) by this quotient, and you will have the dis- 
crepancy between the Gregorian and solar year for 
100,000 years. By this improved method we sup- 
press 25 days, so that the error will only amount to 
25 times 52 minutes. 

SOLUTION. 

100,000-5-4,000 X(l A, 52 m,)=(25 d, 21 h, 40 m.) 
Now (25 d, 21 h, 40 m,)— 25 d,=(21 h, 40 m,) the 
error in 100,000 years. 



30 
CHAPTER III. 



DG VIJSTICAL LETTER. 



Dominical (from the Latin Dominus, lord,) indi- 
cating the Lord's day or Sunday. Dominical letter, 
one of the first seven letters of the alphabet used to 
denote the Sabbath or Lord's day. 

For the sake of greater generality, the days of the 
week are denoted by the first seven letters of the al- 
phabet, A, B, C, D, E, F, G, which are placed in 
the calendar beside the days of the year, so that A 
stands opposite the first day of January, B opposite 
the second, C opposite the third and so on, to G, 
which stands opposite the seventh ; after which A 
returns to the eight, and so on through the 860 days 
of the year. 

Now, if one of the days of the week, Sunday for 
example, is represented by F, Monday will be rep 
resented by G, Tuesday by A, Wednesday by B, 
Thursday by C, Friday by D, and Saturday by E, 
and every Sunday throughout the year will have 
the same character F, every Monday G, every Tues- 
day A, and so with regard to the rest. 

The letter which denotes Sunday is called the 
Dominical or Sunday letter for that year; and when 
the dominical letter of the year is known, the letters 
which respectively correspond to the other days of 
the week become known also. Did the year consist 
of 364 days., or 52 weeks invariably, the first dav of 



31 

the year and the first day of the month, and in fact 
any day of any year, or any month, would always 
commence on the same day of the week. But every 
common year consists of 365 days or 52 weeks and 
one day, so that the following year will begin one 
day later in the week than the year preceding. 
Thus the year 1837 commenced on Sunday, the fol- 
lowing year 1838 on Monday, 1839 on Tuesday, and 
so on. 

As the year consists of 52 weeks and one day, it 
is evident that the day, which begins and ends 
the year must occur 53 times; thus the year 1837 be- 
gins on Sunday and ends on Sanday, so the follow- 
ing year, 1838 must begin on Monday. As A repre- 
sented all the Sunda}^ in 1837, and as A always 
stands for "the first day of January, so in 1838 it will 
represent all the Mondays, and the dominical letter 
goes back from \ to G; so that G represents all the 
Sundays in 1838 5 A all the Mondays, B all the Tues- 
days, and so on, the dominical letter going back one 
place in every year of 365 days. 

While the following year commences one day later 
in the week than the year preceding, the dominical 
letter goes back one place from the preceding year; 
thus while the year 1865 commenced on Sunday, 
1866 on Monday, 1867 on Tuesday, the dominical 
letters are A. G and P respectively. Therefore if 
every year consisted of 365 days, the dominical cyc- 
le would be completed in seven years; so that after 



32 

seven years the first day of the year would again oc- 
cur on the same day of the week. 

But this order is interrupted every four years by 
giving February 29 days, thereby making the year 
to consist of 366 days, which is 52 weeks and two 
days, so that the following year would commence 
two days later in the week than the year preceding, 
thus the year 1888 b^ing leap-year, had two domini- 
cal letters, A andG; A for January and February, 
and G for the rest of the year. The year commenced 
on Sunday and ended on Monday, making 53 Sun 
days and 53 Mondays, and the following year 1889 
to commence on Tuesday. It now becomes evident 
that if the years all consisted of 364 days or 52 
weeks, they would all commence on the same day 
of the week, if they all consisted of 365 days or 52 
weeks and one day, they would all commence one 
day later in the week than the year preceding; i: 
they consisted of 366 days or 52 weeks and two days 
they would commence two days later in the week; 
if 367 days or 52 weeks and three days, then three 
days later and so on, one day later for every additional 
day. It is also evident that every additional day 
causes the dominical letter to go back one place. 
Now in leap-year the 29th day of February is the 
additional or intercalary day. So one letter for Jan- 
uary and February and another for the rest of the 
year. If the number of years in the intercalary period 
were two and seven being the number of days in the 



+* 



33 
week, their product would be 2x7=14;fourteen then, 
would be the number of years in the cycle; again, if 
the number of years in the intercalary period were 
three, and the number of days in the week being 
seven, their product would be 3x7=21; twenty-one 
would then be the number of years in the cycle. But 
the number of years in the intercalary period^ is 
four, and the number of days in the week is seven; 
therefore their product is 4x7=28; twenty-eight is 
then the number of years in the cycle. 

This period is called the Dominical or solar cycle, 
and restores the first day of the year to the same 
day of the week. At the end of the cycle the do- 
mincal letters return again in the same order, on the 
same days of the month. Thus, for the year 1801, 
the domincal letter is B; 1802 C; 1803, B; 1804? A 
and G; and so on, going back five places every four 
years for 28 years; when the cycle being ended, D 
is again dominical letter for 1829, C, for 1830, and 
so on every £8 years forever, according to the Julian 
rule of intercalation. 

But this order is interrupted in the Gregorian 
calendar at the end of the century by the secular 
suppression of the Leap-year, It is not interrupt 
ed, however, at the end of every century, for the 
leap-year is not suppressed in every fourth Centur- 
ial year, consequently the cycle will then be con- 
tinued for two hundred years. It should be here 
stated that this order continued without interrup- 



84 
tion from the commencement of the era until there- 
formation of the calendar in 1582, during which time 
the Julian calendar or Old Style was used. 

It has already been shown that if the number of 
years in the intercalary period be multiplied by 
seven, the number of days in the week, their pro- 
duct will be the number of years in the cycle. Now 
in the Gregorian calendar, the intercalary period is 
400 years; this number being multiplied by seven, 
their product would be 2,800 years, as the interval 
in which the coincidence is restored between the 
days of the year and the days of the week. 

This long period, however, may be reduced to 400 
years; for since the dominical letter goes back five 
places every four years; in 400 years it will go back 
500 places in the Julian and 497 in the Gregorian 
calendar, three intercalations being suppressed in 
the Gregorian every 400 years. Now 497 is exactly 
divisible by seven, the number of days in the week; 
therefore after 400 years, the cycle will be complet- 
ed, and the dominical letters will return again in 
the same order, on the same days of the month. 

In answer to the question, "Why two dominical 
letters for leap-year ?" We reply, because of the 
additional or intercalary day after the 28th of Feb- 
ruary. It has already been shown that every addi- 
tional day causes the dominical letter to go back one 
place. As there are 366 days in leap-year, the let- 
ter must go back two places, one being used for Jan- 



35 

uary and February, and the other for the rest of the 
year. Did we, continue one letter through the year 
and then go back two places, it would cause confus- 
ion in computation, unless the intercalation be made 
at the end of the year. Whenever the intercalation 
is made there must necessarily be a change in the 
dominical letter. Had it been so arranged that the 
additional day was placed after the 30th of June or 
September then the first letter would be used un- 
til the intercalation is made in June or September, 
and the second to me end of the year. Or suppose 
that the end of the year had been fixed as the time 
and place for the intercalation, (which would 
have been much more convenient for com- 
putation), then there would have been no 
use whatever for the second dominical letter, but at 
the end of the year, we would go back two places; 
thus, in the year 1888, instead of A being dominical 
letter for two months merely, it would be continued 
through the year, and then passing back to F, no 
use whatever being made of Gr, and so on at the end 
of every leap-year. Hence it is evident that this 
arrangement would have been much more convenient 
but we have this order of the months, and the num- 
ber of days in the months as Augustus Caesar left 
them eight years before Christ. The dominical letter 
probably was not known until the council of Nice in 
the year of our Lord, 325, where in all probability it 
had its origin. 



36 
CHAPTER IV. 

KULE FOR FINDING THE DOMINICAL LETTER. 



Divide the number of the given year by 4, neglect- 
ing the remainders, and add the quotient to the given 
number. Divide this amount by 7 and if the re- 
mainder be less than three, take it from 3; but if it 
be 3 or more than 3, take it from 10 and the remaind- 
er will be the number of the letter calling A, 1;B, 2; 
C, 3; etc. 

By this rule the dominical letter is found from 
the commencement of the era to October 5th, 1582, 
O. S. From October 15th, 1582 till the year 1700, 
take the remainder as found by the rule from 0, if 
it be less than 6, but if the remainder be 6, take it 
from 13, and so on according to instructions given 
in the table on 41st page. It should be understood 
here, that in leap-year the letter found by the pre- 
ceding rule will be dominical letter for that part 
of the year that follows the 29th of February, while 
the letter which follows it will be the one for Jan- 
uary and February. 

EXAMPLES. 

To find the dominical letter for 1365, we have 
1365-^4=341+ ; 1365+341 = 1706 ; 1706-^7=243, re- 
mainder 5. Then 10—5=5 ; therefore E being the 
fifth letter is the dominical letter for 1365. 

To find the dominical letter for 1620, we have 
1620-j-4=405; 1620+405=2025; -2025--7=289, re- 



37 
mainder 2. Then 6—2=4; therefore D and E are 
the dominical letters for 1620; E for January and 
February and D for the rest of the year. The pro- 
cess of finding the dominical letter is very simple and 
easily understood, if we observe the following order: 

1st. Divide by 4. 

2nd. Add to the given number. 

3rd. Divide by 7. 

4th. Take the remainder from 3 or 10, from the 
commencement of the era to October 5th, 1582. 
From October 15th, 1582 to 1700, from 6 or 13. 
From 1700 to 1800, from 7, and so on. See table on 
41st page. 

We divide by 4 because the intercalary period is 
four years; and as every fourth year contains the 
divisor 4 once more than any of the three preceding 
years, so there is one more added to the fourth year 
than there is to any of the three preceding years ; 
and as every year consists of 52 weeks and one day, 
this additional year gives an additional day to the 
remainder after dividing by 7. For example, the year 

1 of the era consists of 52 w 1 d. 

2 years consist of 104 w 2 d. 

3 years consist of 156 w 3 d. 
(4-=-4)+4=5 years consists of 260 w 5 d. 

Hence the numbers thus formed will be 1, 2, 3, 5, 
6, 7, 8, 10, 11, 12, 13, 15, and so on. 

We divide by 7, because there are seven days in 
the week, and the remainders show how many days 



38 
more than an even number of weeks there are in the 
given year. Take, for example, the first twelve 
years of the era, after being increased by one-fourth 
and we have 

1-1-7=0 remainder 1. Then 3— 1=2=B. 
2-^7=0 " 2 " 3— 2=1=A. 

3-=-7=0 « 3 " 10— 3=7=G. 

5^-7=0 " 5 " 10-5=5=E. P. 

6-f-7=0 " 6 " 10— 6-=-4=D. 

7^7=1 " u 3— 0=3=C. 

8-f-7=l " 1 : < 3— 1=2=B. 

10-=-7=l " 3 " 10— 3=7=G, A.. 

ll-f-7=l " 4 " 10— 4=6=F. 

12-5-7=1 kt 5 " 10— 5=5=E. 

13^7=1 " 6 " l()— 6=4=D. 

15-^7=2 " 1 u 3— 1=2=B, C. 

From this table it may be seen that it is these re- 
mainders representing the number of days more 
than an even number of weeks in the given year, that 
we have to deal with in finding the dominical letter. 
Did the year consist of 364 days, or 52 weeks, in- 
variably, there would be no change in the dominical 
letter from year to year, but the letter that repre- 
sents Sunday in any given year would represent Sun- 
day in every year. Did the year consist of ( nly 
363 days, thus wanting one day of an even 
number of weeks, then these remainders instead of 
being taken from a given remainder, would be add- 
ed to that number, thus removing the dominical let- 



39 
ter forward one place, and the beginning of the year 
instead of being one day later, would be one day 
earlier in the week than in the preceding year. 

Thus', if the year 1 of the era be taken from 3, we 
would have 3 — 1=2; therefore B being the second 
letter, is dominical letter for the year 1 . But if the 
year consist of only 363 days, then, the 1 instead of 
being taken from 3 would be added to 3; then we 
would have 3+1=4; therefore D, being the fourth 
letter would be dominical letter for the year 1. The 
former going back from C to B, the latter forward 
from C to D. 

As seven is the number of days in the week, and 
the object of these subtractions is to remove the do 
minical letter back one place every common year, 
and two in leap-year, why not take these remaind- 
ers from 7? We answer, all depends upon the day 
of the week on which the era commenced. Had Gr, 
the seventh letter been dominical letter for the year 
preceding the era, then these remainders would be 
taken from 7; and 7 would be used until change of 
style in 1582. But we know from computation that 
C, the third letter, is dominical letter for the year 
preceding the era; so we commence with three, and 
take the smaller remainders 1 and 2 from 3; that 
brings us to A We take the larger remainders from 3 
to 6, from 3+7=10. We add the 7 because there are 
seven days in the week. We use the number 10 
until we get back to C, the third letter, the p]ace 



40 
from whence we started. For example, we have 

3— 1=2=B. 
3— 2=1 = A. 
10— 3=7=G. 
10— 4=6=F. 
10— 5=5=E. 
10— 6=4=D. 

3— 0=3=C. 

The cycle of seven days being completed, we com- 
mence with the number three again, and so on until 
1582 when on account of the errors of the Julian 
calendar ten days were suppressed to restore the 
coincidence of the solar and the civil year. Now 
every day suppressed removes the dominical letter 
forward one place; so counting from C to C again is 
seven, D is eight, E is nine, and F is ten. As F is 
the sixth letter, we take the remainders from 1 to 5 
from 6 ; if the remainder be 6, take it from 6+7=13. 
Then 6 or 13 is used till the year 1700, when another 
day being suppressed, the number is increased to 7. 
And again in 1800, for the same reason, a change is 
made to 1 or 8 ; in 1900 to 2 or 9. and so on. It will 
be seen by the table on the 41st page that the small 
er numbers run from 1 to 6 ; the larger ones from 
7 to 13. 

From the commencement of the Christian era to 
October 5th, 1582 take the remainders after dividing 
by 7, from 3 or 10 ; from October 15th 



i 



41 

1582 to 1700 from 6 or 13. 

1700 to 1800 from 7. 

1800 to 1900 from 1 or 8. 

1900 to 2100 from 2 or 9. 

2100 to 2200 from 3 or JO. 

2200 to 2300 from 4 or 11. 

2300 to 2500 from 5 or 12. 

2500 to 2600 from 6 or 13. 

2800 to 2700 from 7 

2700 to 2900 from 1 or 8. 

2900 to 3000 from 2 or 9. 

3000 to 3100 rrom 3 or 10. 

3100 to 3300 from 4 or 11. 

3300 to 3400 from 5 or 12. 

3400 to 3500 from 6 or 13. 

3500 to 3700 from 7. 

3700 to 3800 from 1 or 8. 

3800 to 3900 from 2 or 9. 

3900 to 4000 from 3 or 10. 

4000 to 4100 from 4 or 11. 

4100 to 4200 from 5 or 12. 

4200 to 4300 from 6 or 13. 

4300 to 4500 from 7. 

4500 to 4600 from 1 or 8. 



42 
CHAPTER V. 



RULE FOR FINDING THE DAY OF THE WEEK OF ANY 
GIVEN DATE, FOR BOTH OLD AND NEW STYLES. 



By arranging the dominical letters in the order in 
which the different months commence, the day of 
the week on which any month of any year, or* day 
of the month fall or will fall, from the commence- 
ment of the Christian era to the year of our Lord 
4000 may be calculated. (Appendix (r.) They have 
been arranged thus in the following couplet, in 
which At stands for January, Dover for February, 
Dwells for March, etc. 

At Dover Dwells George Brown, Esquire, 
Good Carlos Finch, and David Fryer. 

Now if A be dominical or Sunday letter for a 
given year, then January and October being repre- 
sented by the same letter, begin on Sunday ; Febru- 
ary, March, and November, for the same reason, 
begin on Wednesday ; April and July on Saturday ; 
May on Monday, June on Thursday, August on Tu~ 
esday, September and December on Friday. It is 
evident that every month in the year must commen- 
ce on some one day of the week represented by one 
of the first seven letters of the alphabet. Now let 
January 1st be represented by A, Sun. 

Feb. 1st (4 w 3 d from the preceding date) by D, Wed. 
Mar.lst4w0d " " " byD, Wed. 

Apr. 1st 4 w 3d " " " by G, Sat. 



43 
May 1st 4w2d " <• ^ by B % Mon. 

June 1st 4 w 3d " " " by E, Thur. 

July 1st 4w2d " " " by G, Sat. 

Aug. 1st 4w3d " <> « byC, Tues„ 

Sep. 1st 4w3d " " " by F, Fri. 

Oct. 1st 4w2d " " " byA 3 Sun. 

ISTov.lst 4w3d " u " by D, Wed. 

Dec, 1st 4 w 2d " " " by F, Fri. 

Now each of these letters placed opposite the 
months respectively represents the day of the week 
on which the month commences, and they are the 
first letters of each word in the preceding couplet. 

To find the day of the week on which a given day 
of any year, will occur we have the following 

eule : 
Find the dominical letter for the year. Read from 
this to the letter which begins the given month, al- 
ways reading from A towards G, calling the domini- 
cal letter Sunday, the next Monday, etc., this will 
show on wJiat day of the week the month commenc- 
ed; then reckoning the number of days from this 
will give the day required. 

EXAMPLES. 

History records the fall of Constantinople on May 
29th, 1453. On what day of the week did it occur ? 
We have then 1453 -f- 4 =363+; 1453+363=1816; 1816 
-7=259, remainder 3. Then 10—3=7; therefore G 
being the seventh letter is dominical letter for 1453.. 
Now reading from Gr to B the letter for May, we 



44 
have G Sunday, A Monday and B Tuesday; hence 
May commenced on Tuesday and the 29th was Tues- 
day. 

The change from Old to New Style was made by 
Pope Gregory XIII, October 5, 1582. On what day 
of the week did it occur { We have then 1582-^-4= 
395+; 1582+395=1977; 1977-=-7=282, remainder 3. 
Then 10—3=7; therefore G being the seventh letter 
is dominical letter for 1582. Now reading from G to 
A the letter for October, we have G Sunday, A Mon- 
day; hence October commenced on Monday, and the 
5th was on Friday. 

On what day of the week did the 15th of the same 
month fall in 1582? We have then 1582-^4=395+; 
1582+395=1977; 1977-^-7 = 282, remainder 3. Then 
6—3=3; therefore C, being the third letter, is the 
dominical letter for 1582. Now reading from C to 
A, the letter for October, we have C Sunday, J) 
Monday, E Tuesday, etc. Hence October corameiv 
ced on Friday, and the 15th was Friday. 

How is this says one? You have just shown by 
computation that October 1582 commenced on Mon 
day, you now say that it occured on Friday. You 
also stated, that the 5th was Friday; you now say 
that the 15th was Friday. This is absured, ten is 
not a multiple of seven, There is nothing absurd 
about it. The former computation was Old Style, 
the latter New Style, the Old being ten days behind 
the New, 



45 

As regards an interval of ten days between the 
two Fridays, there was none; Friday the 5th and 
Friday the 15th was one and the same day; there 
was no interval, nothing ever occured, there was no 
time for anything to occur; the edict of the Pope de- 
cided it; he said the 5th should be called the 15th, 
and it was so. 

Hence to October the 5th 1582 the computation 
should be Old Style; from the 15th, to the end of 
the year New Style. 

On w r hat day of the week did the years 1. 2. and 
3 of the era commence % None of these numbers can 
be divided by 4; neither are they divisible by 7; but 
they may be treated as remainders after dividing by 
seven. Now each of these numbers of years consists 
of an even number of weeks with remainders of 1. 
2. and 3 days respectively. Hence we have then for 
the year 1, 3 — 1=2; therefore B being the second 
letter is the dominical letter for the year 1. Now 
reading from B to A, the letter for January, we have 
B Sunday, C Monday, D Tuesday, etc. Hence Janu- 
ary commenced on Saturday. 

Then we have for the year 2, 3—2=1; therefore 
A being the first letter is dominical letter for the 
year 2; hence it is evident that January commenced 
on Sunday. Again w^e have for the year 3, 10 — 3 
=7;therefore Gr being the seventh letter is dominical 
letter for the year 3. Now reading from G to A, the 
letter for January, we have Gr, Sunday, A Monday; 
hence January commenced on Monday. 



fh 



46 

On what day of the w^ek did the year 4 com- 
mence? Now we have a number that is divisible by 
four; so we have 4-^-4=1; 4+1=5; 5-^-7=0, remain- 
der 5. Then 10 — 5=5; therefore E being the fifth 
letter, is dominical letter for that part of the year 
which follows the 29th of February, while F, the 
letter that follows it, is dominical letter for January 
and February. Now reading from F to A, the letter 
for January, we have F, Sunday, G, Monday, A, 
Tuesday; Hence January commenced on Tuesday 

Now we have disposed of the first four years of 
the era; the dominical letters being B, A, G and 
F, E. Hence it is evident, while one year consists 
of an even number of weeks and one day, two years 
of an even number of weeks and two days, three 
years of an even number of weeks and three days, 
that every fourth year, by intercalation, is made to 
consist of 366 days; so that four years consist of 
an even number of weeks and five days; for we have 
(4-^4)-f-4=5, the dominical letter going back from Gr 
in the year 3, to F, for January and February, and 
from F to E for the rest of year, causing the follow- 
ing year to commence two days later in the week 
than the year preceding. 

The year 1 had 53 Saturdays the year 2, 53 Sun- 
days, the year 3, 53 Mondays, and the year 4. 53 
Tuesdays and 53 Wednesdays, causing the year 5 
to commence on Thursday two days later in the 
week than the preceding year. Now what is true 
concerning the first four years of the era, is true con- 



47 
cerning all the future years, and the reason for the 
divisions, additions and substractions in finding the 
dominical letter is evident. 

The Declaration of Independence was signed July 
4, 1776. On what day of the week did it occur? We 
have then 1776-4=444; 1776+444=2220 ; 2220-7= 
317, remainder 1. Then 7— 1=6, therefore F and G are 
the dominical letters for 1776, G for January and Feb- 
ruary and F for the rest of the year. Now reading 
from F to G, the letter for July, we have F, Sunday, 
G. Monday; hence July commenced on Monday, and 
the fourth was Thursday. On what day of the week 
did Lee surrender to Grant ? which occurred on 
April 9th, 1865. We have then 1865-4=466+; 1865 
+466=2331; 2331—7=333, remainder 0. Then 1—0 
= 1; therefore A being the first letter is dominical 
letter for 1885. Now reading from A to G, the let- 
ter for April, we have A, Sunday, B, Monday, C, 
Tuesday, etc. Hence April commenced on Satur- 
day, and the 9th was Sunday. 

Benjamin Harrison was inaugurated President of 
the United States on Monday, March 4, 1889. On what 
day of the week will the 4th of March fall in 1989 % 
We have then 1989-4=497+; 1989+497=2486; 
2486—7=355, remainder 1. Then 2— 1 = 1; therefore 
A being the first letter, is dominical letter for 1989. 
Now reading from A to D, the letter for March, we 
have A, S mday B, Monday C, Tuesday, and D, 
Wednesday: hence March will commence on Wed- 



48 
nesday, and the 4th will fall on Saturday. Colum- 
bus landed on the island of St. Salvador on Friday, 
October 12, 1492. On what day of the month and 
on what day of the week will the four hundredth 
anniversary fall in 1892 ? 

It is evident that Columbus discovered America 
the 12th of October, Old Style, which may be seen 
by table on the 49th page, to be nine days behind 
the true time; consequently nine days must be added 
to the 12th of October to find the day of the month 
on which the anniversary will fall. We have then 
1892^4=473; 1892+473=2365; 2365—7=337, re- 
mainder 6. Then 8 — 6=2; therefore B and C are 
the dominical letters for 1892, C for January and 
February and B for the rest of the year* Now read- 
ing from B to A the letter for October, we have B 
Sunday, C Monday, etc. Hence October will com- 
mence on Saturday and the 21st will be Friday. 

Although there w T as an error of thirteen days in 
the Julian calendar when it was reformed by Greg- 
ory in 1582, there was a correction made of only 
ten days. There was still an error of three days j 
from the time of Julius Csesar to the council of .Nice 
which remained uncorrected. Gregory restored the 1 
vernal equinox to the 21st of March, its date at the 
meeting of that council, not to the place it occupied ] 
in the time of Cseser, namely the 24th of March. 
Had he done so it would now fall on the 24l1i by 
adopting the Gregorian rule of intercalation. Ap- 
pendix H. 



49 
If desirable calculations may be made in both 
Oldnnd New Styles from the year of our Lord 300. 
There'is no perceptible discrepancy in the Calendars 
however until the close of the 4th Century, when it 
amounts to nearly one day, reckoned in round num- 
bers one day. Now in order to make the calculation, 
proceed according to rule already given for find- 
ing the dominical letter; and for New Style take the 
remainders after dividing by 7 from the numbers in 
table on 49 page. 

From 400 to 500 from 4 

" 500 " 600 " 5 

600 " 700 '• 6 

700 li 900 " 

900 " 1000 " 1 

1000 " 1100 " 2 

1100 " 1300 " 3 

1300 " 1400 " 4 

1400 " 1500 " 5 

1500 " 1700 " 6 

by calculation that from the 



a 



or 11 
" 12 

" 13 

7 

" 8 

" 9 

10 

11 

12 

13 



a 

a 
a 

a 



It will be found 
year 400 to 500 the 



500 

600 

700 

900 

1000 

1100 

1300 

1400 

1500 



u 
u 

* . 

u 

u 
u 
u 
u 



600 
700 
900 
1000 
1100 
1300 
1400 
1500 
1700 



it 

a 
a 
(; 
<< 

a 

a 



discrepancy is 1 day 
2 
3 
4 
5 
6 
7 
8 
9 
10 



U 

u 
u 
a 

u 

u 

u 
u 



u 
u 
u 
u 

t i 
u 
u 
u 
u 



u 
u 
it 

a 

u 
u 
u 



Hence the necessity, in reforming the calender in 



50 

1582, of suppressing ten days. (See table on 51st 
page.) On what day of the week did January com- 
mence in 450 % We have then 450-^4=112+ ; 450+ 
112=562 ; 562-^-7=80, remainder, 2. Then 3—2=1 ; 
therefore, A being the first letter, is dominical letter 
for the year 450, Old Style, and January commenced 
on Sunday. For New Style we have 4 — 2=2 ; there- 
fore B being the second letter is dominical letter for 
the year 450. Now, reading from B to A, the letter 
for January we have B, Sunday ; C, Monday ; D, 
Tuesday ; etc. 

Hence, January commenced on Saturday. Old 
Style, makes Sunday the first day ; New Style makes 
Saturday the first and Sunday the second. On what 
day of the week did January commence in the year 
1250? We have then 1250-^4=312+; 1250+312= 
1562; 1562-^7=223, remainder, 1. Then 3—1=2; 
therefore B, being the second letter, is dominical 
letter for the year 1250, Old Style. Now. reading 
from B to A. the letter for January, we have B, 
Sunday, C, Monday, etc. Hence January commenc- 
ed on Saturday. B is also dominical letter, New 
Style ; for we take the remainder after dividing by 
7, from the same number. 

As both Old and New Styles have the ^ame domin- 
ical letter, so both make January to commence on 
the same day of the week; but Old Style during this 
century is seven days behind the true time; so that 
when it is the first day of January by the Old, it is 
the eight by the New. 



Vernal equinox in the time of Nnma, 

It is here seen that by the errors of the 
Julian Calendar the Vernal equinox is 
made to occur three days earlier 
every 400 years, so that in 
1582, it fell on the 11th 
instead of the 21st 
of March. 



51 
about 700 B. C. 



.March 24 46 B. C. 



23. 



in.. 



12.. 



11 By suppressing 10 days, coincidence Restored in 



Hours behind time, 18 
" 12 
6 
Coincidence, 



in advance 
12 
18 
restored 



Hours behind time, 18 I 6 in advance 

" 12 12 " 
6 1 18 " 
Coincidence, restored 



100 A. 

300 " 
400 " 
500 " 
600 " 
800 " 
900 " 
1000 " 
1200 " 
1300 " 
1400 " 
1600 " 
1700 " 
1800 " 
1900 " 
2000 " 
2100 " 
2200 " 
2300 '« 
2400 " 



By the Georgian rule of intercalation the coincidence of the solar and the civil year is restored very nearly 
every 400 years. 

Appendix I. 



52 
CHAPTER VI. 

A SIMPLE METHOD FOR FINDING THE DAY OF THE 
WEEK OF EVENTS, WHICH OCCUR QUADRENIALLY. 



The inaugural of the Presidents. The day of the 
week on which they have occurred, and on which 
they will occur for the next one hundred years. 
April 30th, 1789, Thursday, George Washington 



March 4th, 


1793, 


Monday, 


U U 


it 


u 


1797, 


Saturday, 


John Adams 


u 


u 


1801, 


Wednesday, Thomas Jefferson. 


u 


u 


1805, 


Monday, 


u u 


4 I 


u 


1809, 


Saturday, 


James Madison. 


i £ 


u 


1813, 


Thursday, 


u u 


U 


u 


1817, 


Tuesday, 


James Monroe. 


U 


c. 


1821, 


Sunday, 


»w U 


u 


u 


1825, 


Friday, 


John Q. Adams. 


(. 


u 


1828, 


Wednesday, Andrew Jackson. 


u 


u 


1833, 


Monday, 


u u 


u 


u 


1837, 


Saturday, 


Martin Van Buren, 


£ I 


" 


1841, 


Thursday, 


Wm. H. Harrison. 


( i 


>' 


1845, 


Tuesday, 


James K. Polk. 


b( 


i. 


1849, 


Sunday, 


Zachary Taylor. 


1 4 


1 4 


1853, 


Friday, 


Frank Pierce. 



1857, Wednesday, James Buchanan. 

J 861, Monday, Abraham Lincoln. 

1865, Saturday, 

1869, Thursday, Ulysses S. Grant, 

1873, Tuesday, 



53 

1877, Sunday, Ruth'f d B. Hayes. 

1881, Friday, James A. Garfield. 

1885, Wednesday, Grove r Cleveland. 

1889, Monday, Benjamin Harrison. 

1893, Saturday, 

1897, Thursday, 

1901, Monday, 

1905, Saturday, 

1909, Thursday, 

1913, Tuesday, 

1917, Sunday, 

1921, Friday, 

1925, Wednesday 

1929, Monday, 

1933, Saturday, 

1937, Thursday, 

1941, Tuesday, 

' 1945, Sunday, 

1949, Friday, 

1953, Wednesday 

1957, Monday, 

1961, Saturday, 

1965, Thursday, 

1969, Tuesday, 

1973, Sunday, 

1977, Friday, 

1981, Wednesday 

1985, Monday, 

1989, Saturday , 
Any one understanding what has been said in a 



54 
preceding chapter concerning the dominical letter, 
can very easily make out such a table without going 
through the process of making calculations for every 
year. As every succeeding year, or any day of the 
year, commences one day later in the week than the 
year preceding, and two days later in leap year, 
which makes live days every four years, and as the 
Presidential term is four years, so every inaugural 
occurs five days later in the week than it did in the 
preceding term. 

Now, as counting forward five days is equivalent 
to counting back two, it will be much more conveni- 
ent to count back tw^o days every term. There is one 
exception, however, to this rule ; the year which 
completes the century is reckoned as a common year, 
(that is three centuries out of four), consequently we 
count forward only four days or back three. 

Commencing, then, with the second inaugural of 
Washington, which occurred on Monday, March 4, 
1793, and counting back two days to Saturday in 
1797, three days to Wednesday in 1801, and two 
days to Monday in 1805, and so on two days every 
term till 1901, when for reasons already given, we 
count back three days again for one term only, alter 
which it will be two days for the next two hundred 
years ; hence anyone can make his calculations as he 
writes, and as fast as he can write. See table on o2d 
page. 



55 

SOME PECULIAKITIES CONCERNING EVENTS WHICH 
EALL ON THE 29th OF FEBRUARY. 

The civil year and the day must be regarded as 
commencing at the same instant. We cannot well 
reckon a fraction of a day, giving to February 28 
days and 6 hours, making the following month to 
commence six hours later every year ; if so then 
March in 

1888 would commence at 6 a. m. 

1889 " " " 12 m. 

1890 " " " 6 p.m. 

1891 ifc " " 12 m. 
again, and so on. 

Instead of doing so, we wait until the fraction ac- 
cumulates to a whole day, then give to February 29 
days, and the year 366. Therefore, events which fall 
on the 29th of February cannot be celebrated annu- 
ally , but only quadrennially ; and at the close 
of those centuries in which the intercalations 
are suppressed only octenniaJly. For example : 
from the year 1696 to 1704, 1796 to 1804, and 
1896 to 1904, there is no 29th day of February ; 
consequently no day of the month in the civil year 
on which an event falling on the 29th of February 
could be celebrated. Therefore, a person born on 
29th of February, 1896, could celebrate no birthday 
till 1904, a period of eight years. 

In every common year February has 29 days, each 
day of the week being contained in the number of 
days in the month four times ; but, in leap year, 
when February has 29 days the day which begins 



56 
and ends the month is contained five times. Let us 
suppose that in a certain year, when February has 
29 days, the month comes in on Friday ; it also must 
necessarily end on Friday. 

After four years it will commence on Wednesday, 
and end on Wednesday, and so on, going back two 
days in the week every four years, until after 28 
years we come back to Friday again. This as has 
already been explained, is the dominical or solar 
cycle. For example : 

The year 4 has five Fridays. 

The year 8 u " Wednesdays. 

The year 12 " " Mondays. 

rp he year 16 " " Saturdays. 

The year 20 " " Thursdays. 

The year 24 wi " Tuesdays. 

Thevear28 " u Sundays. 

The year 32 " '' Fridays. 

So that after 28 years we come back to Friday 
again ; and so on every 28 years, until change of 
style in 1582, Avhen the Georgian rule of intercala- 
tion being adopted by suppressing the intercalations 
in three centurial years out of four interrupts this 
order at the close of these three centuries. For exam- 
ple— 1700, 1800 and 1900. after which the cycle of 28 
years will be continued till 2100, and so on. The cycle 
being interrupted by the Georgian rale of intercala- 
tion, causes all events which occur between 28 and 
12 years of the close of these centuries to fall on the 
same day of the week again in 40 years ; and those 



57 
events, that fall within 12 years of the close of these 
centuries, to fall on the same day of the week again 
in 12 years ; after which the cycle of 28 years will 
be continued during the century. See table on 57th 
page. 

1804 February has five Wednesdays 



1808 
1812 
1816 
1820 
1824 
1828 
1832 
1836 
1840 
1844 
1848 
1852 
1856 
1860 
1864 
1868 
1872 
1876 
1880 
1884 
1888 
1892 
1896 
1900 



Mondays. 

Saturdays. 

Thursdays. 

Tuesdays. 

Sundays. 

Fridays. 

Wednesdays. 

Mondays. 

Saturdays. 

Thursdays. 

Tuesdays. 

Sundays. 

Fridays. 

Wednesdays. 

Mondays. 

Saturdays. 

Thursdays. 

Tuesdays. 

Sundays. 

Fridays. 

Wednesdays 

Mondays. 

Saturdays. 



58 

1904 " " " Mondays. 

1908 " . " " Saturdays. 

1912 " " " Thursdays. 

1916 " " " Tuesdays. 

1920 •< " " Sundays. 

1924 " u " Fridays. 

1928 " " " Wednesdays. 

It will be seen from this table that in 1804 Febru- 
ary had five Wednesdays ; and then again in 1832, 
1860 and in 1888 ; then suppressing the intercalation 
in the year 1900 would make it to occur again in 
1900 or 12 years from the preceding date ; but sup- 
pressing the intercalation suppresses the 29th of 
February ; so opposite 1900 in the table is blank, 
and the 29th of Feburary does not occur till L904, 
and the five Wednesdays do not occur again till 
1928 ; that is 40 years from 1888 when it last occur- 
red. 

Again, taking the five Mondays which occurred 
first in this century in 1808, and then again in 1836, 
1864 and in 1892, you will see, for reasons already 
given, that it will occur again in 12 years, that is 
in 1904 ; and so on with all the days of the week , 
when it will be seen what is peculiar concerning the 
29th of February. 

But attention is particularly called to the five 
Thursdays, which occured first in this century in 
1816, and then again in 1844 and 1872, the last date 
being within 28 years of the close of the century. 
Suppressing the intercalation suppresses the 29th of 



59 
February ; consequently the five Thursdays do not 
occur again till 1912, that is 40 years from the pre- 
ceding date, after which the cycle will be continued 
for two hundred years. 

Hence it may be seen that the dominical or solar 
cycle of 28 years is so interrupted at the close of 
these centuries by the suppression of the leap-year, 
that certain events do not occur again on the same 
day of the week in 40 years ; while others are re- 
peated again on the sam^ day of the week in 12 
years, also the number of years in the cycle, that is, 
28+12=40. 

And again the change of style in 1582, causes all 
events which occur between 28 and 8 years of that 
change, to fall again on the same day of the week 
in 36 years, and all that occur within eight years of 
that change to be repeated again on the same day of 
the week in eight years, after which the cycle of 28 
years is continued for one hundred years ; also, that 
the number of years in the cycle, that is, 28+8=36. 



APPENDIX. 

^->^> 

A.— PAGE 10. 

Authors differ in regard to the length of the Solar 
year. One gives 365 days, 5 hours, 47 minutes and 
51.5 seconds ; another, 365 days, 5 hours, 48 minutes 
and 46 seconds ; and still another, 365 days, 5 hours, 
48 minutes and 49.62 seconds. In this work the last 
has been accepted as the true length of the solar 
year, and a,ll calculations have been made accord- 
ingly. 

B.— PAGE 14. 

Some authors say that the ancient Roman year of 
355 days was increased to 365 by intercalating a 
month of thirty days every three years, so that the 
Romans would have lost nearly one day every four 
years. It is evident that by some means the year 
was too short, and consequently in advance of the 
true or solar time. 

C.-PAGE 18. 

The city w^here the great council was convened in 
325 is not in France as some have supposed, that be- 
ing a more modern city of the same orthography, 
but pronounced Nees. The city which is so fre- 
quently referred to in this work is in Bythinia, one 

60 



61 
of the provinces of Asia Minor, situated about 54 
miles southeast of Constantinople, of the same or- 
thography as the former, but pronounced M-ce, 
and was so named by Lysimachus, a Greek general, 
about 300 years before Christ, in honor of his wife, 
Nicea. 

D.— PAGE 22. 

Sometime during the year 46 B. C, before Caesar 
reformed the old Roman calendar there was inter- 
calated a month of 23 days according to an establish- 
ed method, but still the civil year was in advance 
of the solar year by 67 days ; so that when the earth 
in her annual revolutions should arrive to that point 
of the ecliptic marked the 22d of October, it would 
be the 1st day of January in the Roman year. 

Caesar and his astronomers, knowing this fact, and 
fixing on the 1st day of January, 45 years before 
Christ and 709 from the foundation of Rome, for the 
reformed calendar to take effect, were under the 
necessity of intercalating two months, together, 
consisting of 67 days. Now, as the civil year would 
end on the 22d of October, true or solar time, it 
would be reckoned in the old calendar the 1st day 
of January ; so they let the old calendar come to a 
stand while the earth performs 67 diurnal revolu- 
tions, and thereby restored the concurrence of the 
solar and the civil year. 

As an illustration, let us suppose that in a certain 
shop where hangs a regulator are two clocks to be 
legulated. Bothare set with the regulator at 8 a. m. 



62 
to see how they will run for ten consecutive hours. 
It was found that when it was 6 p. m., by the first 
clock, it was 5:50 by the regulator, the clock having 
gained one minute every hour. 

To rectify this discrepancy we must intercalate 
ten minutes by stopping the clock until it is six by 
the regulator. By thes means the coincidence is re- 
stored, and the time lost in the preceding hours is 
now reckoned in this last hour making it to consist of 
70 minutes. By this it may be seen how Csesar re- 
formed the Roman calendar. The Roman year was 
too short, by reason of which the calendar was 
thrown into confusion, being 90 days in advance of 
the true time, so that December, January, and Feb- 
ruary,- took the place in the seasons, of September, 
October, and November; and September, October 
and November, the place of June, July and August. 
To make the correction he must stop the old Roman 
clock (the calendar) while the Earth performs 90 
diurnal revolutions to restore the concurrence of 
the solar and the civil years, making the year 46 
B, C, to consist of 445 days. 

It was also found that when it was 6 p. m., by the 
regulator, it was only 5:50 by the second clock, it 
having lost one minute every hour. To rectify this 
discrepancy we must suppress ten minutes, calling 
it 6 p. m., turning the hands of the clock to coincide 
with the regulator, making the last hour to consist 
of only 50 minutes, too much time having been reck- 
oned in the preceding hours. It may be seen by 



63 
this illustration, how Gregory corrected the Julian 
Calendar, the Julian year was too long, consequent- 
ly behind true or solar time, so that when the cor- 
rection was made in 1582, the ten days gained had 
to be suppressed to restore the coincidence, making 
the year to consist of only 355 days. 

As the solar year consists of 365 days and a frac- 
tion, Caesar intended to retain the concurrence of 
the solar and the civil year by intercalating a day 
every four years ; but this made the year a little too 
long, by reason of which it became necessary, in 
1582, to rectify the error, and by adopting the Gre- 
gorian rule, three intercalations are suppressed 
every 400 years ; so that by a series of intercalations 
and suppressions, our calendar may be preserved in 
its present state of perfection. 

E.— PAGE 22. 

As the day and the civil year always commence at 
the same instant, so they must end at the same in- 
stant; and as the solar year always ends with a 
fraction, not only of a day, but of an hour, a min- 
ute and even a second ; so there is no rule of inter- 
calation by which the solar and the civil year can 
be made to coincide exactly. But the discrepancy 
is only a few hours in a hundred years, and that is 
so corrected by the Gregorian rule of intercalation 
that it would amount to a little more than a day in 
4,000 years ; and by the improved method less than 
a day in 100,000 years. 

* P.— PAGE 25. 

It has been stated that by adopting the Julian 
rule of intercalation, time was gained ; it has also 



64 
been stated that by the same rule time was lost. 
Now both are true. Time is gained in that there is 
too much time in a given year, in other words the 
year is too long ; but what is gained in a given year 
is lost to the following years. 

As an illustration let us take the case of the sup- 
posed solar year of 365 days, and the civil year of 
366. The civil year would gain one day every year, 
or be too long by one day ; but the one day gained 
is lost to the following years, and if continued 31 
years, when the Earth is in that part of its orbit 
marked the 1st day of January 32, the civil year 
would reckon the 1st day of December 31 ; so that in 
the thirty-one years would reckon thirty one days 
too much, and before the civil year is completed, the 
Earth will have passed on in its orbit to a point 
marked the 1st day of February. 

Now to reform such a calendar, we would have to 
suppress or drop the thirty-one days, by calling the 
1st day of December the 1st day of January, and 
thus the month of December would disappear from 
the calendar in the year 31, making a year of only 
eleven months, consisting of 334 days. 

If this method be continued 92 years, there would 
be gained 92 days, to the loss of 92 days in the year 
92. If the calendar be now reformed by suppress- 
ing 92 days, calling the first day of October 92, the 
first day of January 93, then October, November, 
aud December would disappear from the calendar in 
the year 92 ; and if continued 365 years there would 
be crowded into 364 years, 364 days too much ; gain- 
ed to the 364 years to the total loss of the year 365, 



65 
passing from 364 to 366 ; 365 disappearing from the 
calendar. 

G.— PAGE 43. 

An era is a fixed point of time from which a serie s 
of years is reckoned. Among the nations of the 
Earth there are no less than twenty-five different eras; 
but the most of them are not of enough importance 
to be mentioned here. Attention is particularly 
called to the Roman era which commenced with the 
building of the city of Rome 753 years before Christ. 

Also the Mahometan era, or era of the Hegira, em- 
ployed in Turkey, Persia, and Arabia, which is dated 
from the flight of Mahotnet from Mecca to Medina, 
which was Thursday night, the 15th of Ju?y, A. D., 
622, and it commenced on Friday the day following. 

But there is a point from which all computation 
originally commenced, namely the creation of man. 
Such an era is called the Mundane era. How -there 
are different Mundane eras, — the common Muudane 
era 4,004 B. C, the Grecian Mundane era 5,598 B. 
C, and the Jewish Mundane era 3,761 B. C. All 
these commence computation from the same point;, 
but differ in regard to the time which has elapsed 
since their computation commenced. God's people 
used the Mundane era, until the great .. Creator ap- 
peared among us, as one of us, in the person of our 
Lord Jesus Christ accomplish the great w^ork 
of redemption; then his name was introduced as the 
turning point of the ages, the starting point of com- 
putation. 



66 

This was done by Dionysius Exiguns in the year 
of our Lord about 540, known at that time as the 
Dionysian, as well as the Christian era, and was 
first used in historical works by the venerable Bede 
early in the 8th century "It was a great thought 
of the little monk (whether so called from his hu- 
mility or littleness of stature is unknown), to view 
Christ as the turniug point of the ages, and to intro- 
duce this view into chronology.' 5 

All honor to him who introduced it, and to the 
nations which have approved, for thus honoring the 
great Redeemer. Dionysius x>i obably did not know, 
neither is it now known for a certainty the year of 
Christ's birth, but it is evident, however, from the 
best authorities, that the era commenced at least five 
years too late, and probably more 
H. -PAGE 48. 

It is recorded that, in the time of Numa, the ver- 
nal equinox fell on the 25th of March, and that Jul- 
ius Caesar restored it to the 25th, when he reformed 
the ancient Roman calendar in the year 46 B. C. It 
is also recorded that in less than 400 years 
from that time, at the meeting of the Council of 
Nice in 325, it had fallen back to the 21st, — four 
days in less than 400 years. 

Now there is an error somewhere, for it is found 
by actual computation that the discrepancy between 
the solar and the Julian year is about three days in 
400 years. It certainly is true that the vernal equi- 
nox fell on the 21st in 325, and was restored to that 
place by Gregory in 1582; since which time it has 



67 
been made to fall on the 21st by the Gregorian rule 
of intercalation. Now with these facts before us, 
we must come to the conclusion that the vernal equi- 
nox did not fall on the 25th of March in the time of 
Numa, nor of Julius Caesar, but the 24th. 
I.— PAGE 51. 

The concurrence of the solar and the civil year 
was restored by Gregory in 1582, or 1600 is the 
same in computation; but the discrepancy between 
civil and solar time is 11 minutes and 10.38 seconds 
every year, which in one hundred years will amount 
to 18 hours and 37.3 minutes; reckoned in round 
numbers 18 hours, and is represented on the Chart, 
Hours behind time 18. 

The intercalary day or 24 hours being suppressed 
in 1700, causes the civil year to be 6 hours in advance 
of the solar, and is re j resented on the chart 6 hours 
in advance. 

Now this discrepancy of 18 hours for tlie next 100 
years, will cause the civil year in 1800 to be 12 hours 
behind; again suppressing the intercalation it will 
be 12 hours in advance. In 1900 it will be 6 hours 
behind, but the correction makes 18 hours 
in advance. The 18 hours gained the next 
one hundred years restores the coincidence in the 
year 2000, and so on, the solar and the civil year being- 
made to coincide very nearly every 400 years. 

From close examination it will become evident that 
the solar and the civil year coincide twice every 400 



68 
years, though no account is made of it in computa- 
tion. From 6 hours in advance in 1700, the civil 
year falls back to 12 hours behind the solar in 1800, 
consequently they must coincide in 1733. 

Again from 12 hours in advance in 1800, it falls 
back to 6 hour? behind the solar in 1900, conse- 
quently they must coincide again in 1867. 

Discrepancy between Julian and solar time in — 
1 year is (365 d, 6 h.)— (365 d, 5 h, 48 m, 49.62 s)= 
(11 m, 10.38 s.) 

100 years is (11 m, 10.38 s.)Xl00=(18 h, 37.3 id.) 
400 " " (18 h, 37.3 m.)X4=(3 d, 2 h, 29.2m.) 

4,000 *' (3d, 2 h, 29.2 m.)XlO=(31 d, h, 52m.) 
100,000 u (31 d, Oh, 52 m.)X25 = (775 d, 21 h, 40 m.) 

Discrepancy between Gregorian and solar time in — 
1 year is --------- - .373 m. 

100 years is .373 m. xl00== . - - - - 37.3 m. 

400 '' 37.3 m. X4= - - - 2 h, 29.2 m. 

4,000 " (2 h, 29.2m.)xl0= 1 d, h' 52 m. 

100,000 " (1 d, h, 52 m.)x25=25 d, 21 h, 40 m. 

Discrepancy between corrected Gregorian and so- 
lar time in — 

4,000 years is (1 d, h, 52 m)— 1 dav = - 52 m- 
100,000 - " (52 m. X 25 )= 21 h. 40 



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022 008 929 




